Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The distribution of the largest digit for parabolic Iterated Function Systems of the interval

Hiroki Takahasi

TL;DR

This work extends the study of growth rates of the largest digits in continued-fraction-type expansions to a broad class of infinite parabolic IFS on the unit interval. By combining a finite-IFS approximation strategy with a digit-insertion construction, the authors show that for a $d$-decaying parabolic IFS satisfying (B1)–(B2), every level set $L(\alpha)$ has the same Hausdorff dimension as the limit set $\Lambda(\Phi)$, i.e., $\dim_{H}L(\alpha)=\dim_{H}\Lambda(\Phi)$ for all $\alpha\in[0,\infty]$. The two-step approach—approximating by finite IFSs and embedding large digits without distortion—extends Wu–Xu’s regular-continued-fraction result to backward and related continued fractions and beyond, under a robust dimension-theoretic framework for non-uniformly expanding Markov maps with infinitely many branches. The results yield a flexible method to analyze digit-growth phenomena in a wide range of infinite-IFS settings, with practical implications for parabolic dynamics and associated continued-fraction representations.

Abstract

We investigate the distribution of the largest digit for a wide class of infinite parabolic Iterated Function Systems (IFSs) of the unit interval. Due to the recurrence to parabolic (neutral) fixed points, the dimension analysis of these systems become more delicate than that of uniformly contracting IFSs. We show that the Hausdorff dimensions of level sets associated with the largest digits are constantly equal to the Hausdorff dimension of the limit set of the IFS. This result is an analogue of Wu and Xu's theorem [Math. Proc. Camb. Phil. Soc. {\bf 146} (2009), 207--212] on the regular continued fraction. Examples of application of our result include the backward (aka minus, or negative) continued fractions, even integer continued fractions, and go beyond. Our main technical tool is a dimension theory for non-uniformly expanding Markov interval maps with infinitely many branches.

The distribution of the largest digit for parabolic Iterated Function Systems of the interval

TL;DR

This work extends the study of growth rates of the largest digits in continued-fraction-type expansions to a broad class of infinite parabolic IFS on the unit interval. By combining a finite-IFS approximation strategy with a digit-insertion construction, the authors show that for a -decaying parabolic IFS satisfying (B1)–(B2), every level set has the same Hausdorff dimension as the limit set , i.e., for all . The two-step approach—approximating by finite IFSs and embedding large digits without distortion—extends Wu–Xu’s regular-continued-fraction result to backward and related continued fractions and beyond, under a robust dimension-theoretic framework for non-uniformly expanding Markov maps with infinitely many branches. The results yield a flexible method to analyze digit-growth phenomena in a wide range of infinite-IFS settings, with practical implications for parabolic dynamics and associated continued-fraction representations.

Abstract

We investigate the distribution of the largest digit for a wide class of infinite parabolic Iterated Function Systems (IFSs) of the unit interval. Due to the recurrence to parabolic (neutral) fixed points, the dimension analysis of these systems become more delicate than that of uniformly contracting IFSs. We show that the Hausdorff dimensions of level sets associated with the largest digits are constantly equal to the Hausdorff dimension of the limit set of the IFS. This result is an analogue of Wu and Xu's theorem [Math. Proc. Camb. Phil. Soc. {\bf 146} (2009), 207--212] on the regular continued fraction. Examples of application of our result include the backward (aka minus, or negative) continued fractions, even integer continued fractions, and go beyond. Our main technical tool is a dimension theory for non-uniformly expanding Markov interval maps with infinitely many branches.

Paper Structure

This paper contains 13 sections, 11 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Saturation and mild distortion
  4. Entropy, dimension, decay of fundamental intervals
  5. Construction of finite IFSs with large limit sets
  6. Proof of the main result
  7. Initial setup
  8. Construction of a subset of the level set
  9. Completing the proof of Theorem \ref{['mainthm']}
  10. The first technical lemma
  11. The second technical lemma
  12. Proof of Proposition \ref{['prop-lip']}
  13. Verification of (B1) (B2) and examples

Key Result

Theorem 1.1

Let $d>1$, and let $\Phi=\{\phi_i\}_{i\in I}$ be a $d$-decaying parabolic IFS on $[0,1]$ that satisfies (B1) (B2). For any $\alpha\in[0,\infty]$ we have

Theorems & Definitions (19)

  • Theorem 1.1
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 3.1
  • proof
  • ...and 9 more